بِسْمِ ٱللَّٰهِ ٱلرَّحْمَٰنِ ٱلرَّحِيمِ
In the name of Allah, the Most Gracious, the Most Merciful

Chapter 4: Sequences and Series

Unit 4: Complete Solved Notes

📚 Core Definitions

🧮 Essential Formulas

Formula Name Expression
General Term of Arithmetic Sequence
Sum of Terms of Arithmetic Series or
General Term of Geometric Sequence
Sum of Terms of Geometric Series for
Sum of Infinite Geometric Series for
Arithmetic Mean
Geometric Mean
Harmonic Mean
Triangular Sequence
Pascal Sequence
Relation Between A.M, G.M, H.M
Recurring Decimal to Fraction

✍️ Guess-Paper Questions (Solved)

📝 Exercise 4.1

Q4. Find the values of nth term of triangular sequence when and .
Q6. Find the Pascal sequence when and .

Solution:

The Pascal sequence for the th row is given by for .

(i) For :

, , , , ,

Sequence:

(ii) For :

, , , , , ,

Sequence:

(iii) For :

, , , , , , ,

Sequence:

(iv) For :

, , , , , , , ,

Sequence:

📝 Exercise 4.2

Q5. Find the 21st term of the A.P if its 6th term is 11 and the 15th term is 47.
Q6. (a) Which term of the sequence is 75?
(b) Which term of the sequence is ?
(c) Which term of the sequence is 148?
Q9. If are in A.P, show that .
Q11. An object falling from rest, falls 12 metres during the first second, 24 metres during the next second, 36 metres during the third second and so on. How much will it fall during the 8th second?
Q12. A man deposits Rs. 13,000 in a bank in the first month; Rs. 14,500 in the second month; Rs. 16,000 in third month and so on. Find how much he has to deposit in the bank at the end of a year.
Q13. A boy saves Rs. 200 at the end of the first week and goes on increasing his saving for Rs. 25 weekly. After how many weeks, his weekly saving will be Rs. 2000.

📝 Exercise 4.3

Q1(i). Find the A.M between 18 and 26.
Q2. Insert three A.Ms. between 3 and 11.
Q5. Insert 6 A.Ms. between and .
Q6. If 5 and 8 are two A.Ms. between and , find and .
Q7. Find so that may be the A.M. between and .

📝 Exercise 4.4

Q4. If , then find the series.
Q5. Find the sum of first 100 natural numbers which are neither exactly divisible by 3 nor by 7.
Q7. The sum of terms of two arithmetic series are in the ratio . Find the ratio of their 8th terms.
Q8. The sum of three numbers in A.P. is 27 and their product is 405. Find the numbers.
Q9. Find five numbers in A.P. whose sum is 25 and the sum of whose squares is 135.
Q10. The sum of Rs. 42,000 is distributed among five persons so that each person after the first receives Rs. 80 less than the preceding person. How much does each person receive?
Q11. A well digging company charges Rs. 1,250 for the first meter, Rs. 1500 for the second meter and Rs. 1,750 for the third meter and so on. What is the depth of a well that costs Rs. 50,000.
Q12. A man borrows Rs. 25,000 and agrees to repay with a total profit of Rs. 10,000 in 10 installments, each installment being less than the preceding by Rs. 200. What should be his first installment?

📝 Exercise 4.5

Q2. In a G.P, , . Find its th term.
Q3. Find the th term of the geometric sequence if: and .
Q5. How many terms are in G.P. ?
Q6. Find three consecutive numbers in G.P. whose sum is 39 and their product is 729.
Q7. The number of bacteria in a culture increased in G.P from 515,000 to 15,45,000 in 7 days. Find the daily rate of increase, assuming the rate of increase to be constant.
Q8. Find the profit on Rs. 1000 for 5 years at 4% per annum compound profit.

📝 Exercise 4.6

Q2. Insert:
(i) Two G.Ms between 3 and 81.
(ii) Three G.Ms between 2 and .
Q3. The A.M. between two numbers is 29 and the geometric mean is 21, find the numbers.
Q4. For what value of , is the G.M. between and ?
Q5. Show that the th root of the product of geometric means between and is the geometric mean between and .
Q6. The A.M. of two positive integral numbers exceeds their (positive) G.M by 2 and their sum is 20. Find the numbers.
Q7. The A.M. between two numbers is 5 and their (positive) G.M. is 4. Find the numbers.

📝 Exercise 4.7

Q1(iv). Find the sum: to terms.
Q1(v). Find the sum: to terms.
Q1(vi). Find the sum: to terms.
Q1(vii). Find the sum: to terms.
Q2(ii). Find the sum to terms of the series: , where and are proper fractions.
Q3. If then find its sum up to terms.
Q4. Find the sum of the following infinite geometric series:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
Q5. Find the Vulgar fraction equivalent to the following recurring decimals:
(i)
(ii)
(iii)
(iv)
(v)
Q6. If if , then prove that .
Q7. The sum of an infinite geometric series is half the sum of the squares of its terms. If the sum of its first two terms is , find the series.
Q8. Joining the midpoints of the sides of an equilateral triangle, an equilateral triangle having half the perimeter of the original triangle is obtained. We form a sequence of nested equilateral triangles in this manner with the original triangle having perimeter . What will be the total perimeter of all the triangles formed in this way?

📝 Exercise 4.8

Q2. If the 7th and 10th terms of an H.P are and respectively, find its 14th and 20th terms.
Q3. The first term of H.P is and fifth term is . Find its 9th term.
Q4. If the th term of an H.P. is , the th term is : prove that the th term is .
Q5. If the th term of an H.P. is , the th term is : prove that the th term is .

📝 Exercise 4.9

Q1(i). Insert the H.M between and .
Q2(i). Insert four H.Ms between and .
Q4. Prove that the square of the geometric mean of two numbers equals the product of the A.M and the H.M of the two numbers.
Q5. Find so that may be H.M. between and .
Q7. Find the arithmetic, harmonic and geometric means between 1 and 9. Also verify that .
Q8. The A.M of two numbers is 8 and H.M is 6. Find the numbers.
Q9. The H.M. of two numbers is and G.M. is 6. Find the numbers.

📝 Review Exercise 4

Q3. There are A.Ms. between 8 and 32 such that the ratio of the third and 7th means is 3:5, find the value of .
Q5. How many terms are there in a G.P if , and ?
Q8. If A.M and H.M between two numbers are 5 and respectively. Find the numbers.
Q9. If G.M. and H.M. between two numbers are 15 and respectively. Find the numbers.
Q10. The second term of an H.P is and the fifth term is . Find the 12th term.

❓ Multiple Choice Questions (Interactive)

1. A sequence is a function whose domain is set of:
(a) integers
(b) rational numbers
(c) natural numbers
(d) real numbers
2. If H is the harmonic mean between and then H is:
(a)
(b)
(c)
(d)
3. If and then :
(a) 3
(b) 5
(c) 14
(d) 20
4. A sequence in which is the same number for all is called:
(a) A.P
(b) G.P
(c) H.P
(d) None of these
5. If are in A.P, then is called:
(a) A.M
(b) G.M
(c) H.M
(d) Mid-point
6. Arithmetic mean between and is:
(a)
(b)
(c)
(d)
7. The harmonic mean between and is:
(a)
(b)
(c)
(d) None of these
8. For any G.P the common ratio is equal to:
(a)
(b)
(c)
(d) for
9. No term of a G.P is:
(a) 0
(b) 1
(c) Negative
(d) Imaginary number
10. The sum of infinite geometric series is a finite number if:
(a)
(b)
(c)
(d)
11. If the reciprocals of the terms of a sequence form an A.P, then it is:
(a) Harmonic sequence
(b) Arithmetic sequence
(c) Reciprocal sequence
(d) Series
12. If is A.M between & , then is equal to:
(a) 0
(b) -1
(c) 1
(d)
13. If are in A.P., then
(a) 0
(b) 1
(c) 2
(d) 3
14. G.M between and 8 is:
(a) or
(b) 4 or
(c) 16 or
(d) 3 or
15. General term of a sequence is . Its term is:
(a) -4
(b) -16
(c) 16
(d) 4
16. If then infinite geometric series is \text{-----------}:
(a) Convergent
(b) Divergent
(c) Undefined
(d) both a and b
17. The harmonic mean of and is:
(a)
(b)
(c)
(d)
18. If , then and are in:
(a) A.P
(b) G.P
(c) H.P
(d) None of these

🔑 MCQ Answer Key

1. C
2. C
3. D
4. A
5. A
6. A
7. D
8. C
9. A
10. D
11. A
12. A
13. D
14. A
15. B
16. B
17. A
18. B